INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 15 (2003) S3411–S3420
PII: S0953-8984(03)66420-9
Capillary condensation of colloid–polymer mixtures
confined between parallel plates
Matthias Schmidt1 , Andrea Fortini and Marjolein Dijkstra
Soft Condensed Matter, Debye Institute, Utrecht University, Princetonplein 5,
3584 CC Utrecht, The Netherlands
Received 21 July 2003
Published 20 November 2003
Online at stacks.iop.org/JPhysCM/15/S3411
Abstract
We investigate the fluid–fluid demixing phase transition of the Asakura–Oosawa
model colloid–polymer mixture confined between two smooth parallel hard
walls using density functional theory and computer simulations. Comparing
fluid density profiles for statepoints away from colloidal gas–liquid coexistence
yields good agreement of the theoretical results with simulation data.
Theoretical and simulation results predict consistently a shift of the demixing
binodal and the critical point towards higher polymer reservoir packing fraction
and towards higher colloid fugacities upon decreasing the plate separation
distance. This implies capillary condensation of the colloid liquid phase, which
should be experimentally observable inside slitlike micropores in contact with
a bulk colloidal gas.
1. Introduction
Capillary condensation denotes the phenomenon that spatial confinement can stabilize a liquid
phase coexisting with its vapour in bulk [1, 2]. In order for this to happen the attractive
interaction between the confining walls and the fluid particles needs to be sufficiently strong.
Although the main body of work on this subject has been done in the context of confined
simple liquids, one might expect that capillary condensation is particularly well suited to be
studied with colloidal dispersions. In these complex fluids length scales are on the micron
rather than on the ångström scale which is typical for atomic substances. Attraction between
colloidal particles can be generated and precisely tuned by adding non-adsorbing polymer.
The presence of the polymers induces an effective attraction between the colloids that may
lead to fluid–fluid phase separation reminiscent of the gas–liquid transition in atomic systems:
the phase that is poor in colloids corresponds to the gas and the phase that is dense in colloids
corresponds to the liquid [3].
1 On leave from Institut für Theoretische Physik II, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1,
D-40225 Düsseldorf, Germany.
0953-8984/03/483411+10$30.00
© 2003 IOP Publishing Ltd
Printed in the UK
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M Schmidt et al
A particularly simple model to study colloid–polymer mixtures is that proposed by Asakura
and Oosawa (AO) [4] and Vrij [5], where colloids are represented as hard spheres and polymers
as effective, overlapping spheres that cannot penetrate the colloids. Indeed this model displays
stable gas and liquid phases and also a crystalline solid [6–12].
The same mechanism that generates an effective attraction between two colloidal particles
via depletion of polymers gives also rise to attraction between a single colloidal particle and a
simple hard wall leading to strong adsorption of colloids at the wall [13]. Close to the gas branch
of the demixing binodal the hard wall is wet by the colloidal liquid, purely driven by entropy,
and an intriguing scenario of entropic wetting with a finite sequence of layering transitions
in the partial wetting regime was found with density functional theory (DFT) [14, 15] and
computer simulations [11], and recently also in a rod–sphere mixture [16]. In fact, there
is also experimental evidence for wetting of a smooth hard substrate by the colloid-rich
phase [17–19]. Less attention has been paid to the influence of strong confinement on colloid–
polymer mixtures, exceptions being exposure to a standing laser field [20] and immersion into
random sphere matrices acting as porous media [21, 22].
In this work we investigate the effect of contact with two narrowly spaced, parallel
smooth hard walls. We use Monte Carlo (MC) computer simulation and DFT to investigate
a range of wall separation distances from ten down to one colloid diameter—covering the
dimensional crossover to two remaining space coordinates. Both theory and simulation
treat the polymers explicitly, and hence include, in principle, all polymer-induced manybody interactions between colloids. Testing the theory for a case of strong confinement by
comparing colloid and polymer density profiles away from coexistence with simulation data
demonstrates good accuracy. From theory and simulation results for fluid–fluid coexistence we
find that indeed confinement stabilizes the colloidal liquid inside the capillary, hence capillary
condensation does occur. These findings should be experimentally observable in colloid–
polymer mixtures prepared such that the colloidal bulk gas is in contact with a thin slit pore,
and the adsorption inside the pore is measured.
The paper is organized as follows. In section 2 we define the AO model between parallel
walls explicitly. In section 3 both the MC and DFT techniques are presented. Section 4
discusses results for one-body density profiles and the demixing phase diagram. We conclude
in section 5.
2. Model
The AO model is a binary mixture of colloidal hard spheres (species c) of diameter σc and of
spheres representing polymer coils (species p) with diameter σp [4, 5]. The pair interaction
between colloids is that of hard spheres: Vcc (r ) = ∞ if r < σc and zero otherwise, where r
is the separation distance between particle centres. The interaction between a colloid and a
polymer is also that of hard spheres: Vcp (r ) = ∞ if r < (σc + σp )/2 and zero otherwise. The
polymers, however, are assumed to be ideal, hence the polymer–polymer interaction vanishes
for all distances, Vpp (r ) = 0. To model the confinement between smooth parallel hard walls
we consider external potentials acting on species i = c, p, given by
0
σi /2 < z < H − (σi /2)
Vext,i (z) =
(1)
∞
otherwise,
where z is the spatial coordinate normal to the walls, and H is the wall separation distance;
see figure 1 for an illustration of the situation.
Capillary condensation of colloid–polymer mixtures confined between parallel plates
S3413
z
Figure 1. Illustration of the AO model of hard sphere colloids (grey circles) of diameter σc and
ideal polymers (transparent circles) of diameter σp confined between parallel hard walls of area A
and separation distance H . Colloids behave as hard spheres, polymers cannot penetrate colloids,
and polymers may freely overlap. The walls are impenetrable to both components. The z-axis is
perpendicular to the walls, and the origin is located at the lower wall.
The size ratio q = σp /σc is a geometric control parameter; the ratio H /σc rules the strength
of the confinement. We denote the packing fractions by ηi = πσi3 Ni /(6 AH ), where Ni is
the number of particles of species i and A is the lateral (normal to the z-direction) area of
the system. As alternative thermodynamic parameters, we use the packing fraction in a pure
reservoir of polymers, ηpr , and the fugacity of the colloids, z c .
3. Methods
3.1. Computer simulations
We employ Gibbs ensemble MC simulations, where phase coexistence is directly accessible
through the use of two simulation boxes, each representing one of the coexisting states [23, 24].
Both boxes contain the confining walls at fixed plate separation distance H . Let Ni,a be the
number of particles of species i in box a = 1, 2, and Aa H be the volume of box a, where Aa is
its lateral area. Then A1 H + A2 H = constant and Ni,1 + Ni,2 = constant, while fluctuations of
Ni,a and Aa inside each individual box a = 1, 2 are allowed. Thus three different (randomly
chosen) types of MC move are performed: single-particle moves of colloids and polymers
inside each simulation box, moves that transfer particles from one box to the other to ensure
equal chemical potential, and volume exchanges between both boxes to ensure equal wall–fluid
interfacial tension via rescaling the lateral box dimensions (and the x- and y-components of
particle positions) while keeping the plate separation distance H (and the z-components of
all particle positions) constant. We start from a random, non-overlapping configuration, and
use 108 MC moves for equilibration and typically 3 × 108 production moves for each pair of
coexistence statepoints. Acceptance ratios for particle exchange depend strongly on statepoint
and decrease, upon increasing ηpr , from about 50% close to the critical point to about 5%
for the highest values considered. The coexistence densities are obtained from the maxima
of a histogram of the number of particles in each box. Fugacities are calculated via the test
particle insertion method [25]. Typical total particle numbers are Nc,1 + Nc,2 = 200–400
for colloids and, dependent on the statepoint, Np,1 + Np,2 = 600–2000 for polymers. The
total lateral area of both boxes, A1 + A2 , is between 400σc2 and 2500σc2 . Each run was
divided into ten blocks and error bars were obtained from one standard deviation of the block
averages.
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M Schmidt et al
3.2. Density functional theory
The grand potential of the binary mixture is written as a functional of the (inhomogeneous)
one-body density distributions ρi (r) as
[ρc , ρp ] = kB T
dr ρi [ln(ρi (r)3i ) − 1]
i=c,p
+ Fexc [ρc , ρp ] +
dr ρi (r)[Vext,i (r) − µi ],
(2)
i=c,p
where i is the thermal wavelength and µi is the chemical potential of species i , kB is
Boltzmann’s constant and T is temperature. Although equation (2) is general, for the
present application we take the external potentials, Vext,i , to be those modelling the slit pore,
equation (1). The first term on the right-hand side of (2) is the free energy functional of a
binary ideal gas; the unknown part in (2) is the (Helmholtz) excess free energy functional,
Fexc , that describes the influence of interparticle interactions. Here we take the fundamental
measures approximation [26] of [9, 10] that is specifically tailored for the AO model. This is
a weighted density approximation where Fexc is expressed as a spatial integral over an excess
free energy density, which depends on a set of weighted densities that are obtained through
convolutions of the bare density profiles with geometrically inspired weight functions. Details
can be found in [9, 10]. In order to obtain density profiles the minimization conditions,
δ
= 0,
δρi (r)
i = c, p,
(3)
are solved by a standard iteration technique, and the value of the grand potential is obtained
by inserting the solutions of (3) into (2). In order to have direct access to phase coexistence
we minimize two systems simultaneously and adjust the chemical potentials of both species
iteratively such that in the final state both systems possess the same grand potential. This
procedure is inspired by the Gibbs ensemble MC method above.
4. Results
In the following we will restrict ourselves to the polymer-to-colloid size ratios q = 1. For this
value considerable knowledge about bulk [11] and hard wall behaviour [11, 15] is available.
Moreover, it is an experimentally readily accessible case.
4.1. Comparison
We start by comparing density profiles for statepoints where we expect the system to be in
the one-phase mixed fluid state. As our DFT is known to be accurate in bulk [10], and at
single hard planar walls [15], we focus on a case of strong confinement and hence consider
H /σc = 2. Figure 2 displays colloid and polymer profiles as a function of the z-coordinate
normal to the walls. Both colloid and polymer density profiles are largest at contact with the
walls, z/σc = 0.5, 1.5.
The agreement of the theoretical curves with the simulation data is quantitatively good. We
note, however, that for still larger values of ηpr differences begin to emerge (results not shown).
As such conditions get close to the demixing transition, we attribute this to the differences in
the results for the demixing binodal.
Capillary condensation of colloid–polymer mixtures confined between parallel plates
S3415
0.7
c, state 2
0.6
0.4
3
ρ i σ i π /6
0.5
c, state 1
0.3
0.2
p, state 1
p, state 2
0.1
0
0
0.5
1
1.5
2
z/σ c
Figure 2. Density profiles ρi (z)σi3 π/6, i = c, p as a function of the coordinate z/σc normal to the
plates for plate separation distance H /σc = 2. Results from DFT (curves) and MC (symbols) are
shown for ηc = 0.0997, ηpr = 0.266 (statepoint 1) and ηc = 0.157, ηpr = 0.367 (statepoint 2).
4.2. Phase diagram
For large enough concentration of polymer, phase separation into colloid-poor gas and colloidrich liquid is expected to occur in our model capillary. As an illustration we display snapshots
of a pair of coexisting states in figure 3 for large wall separation distance H /σc = 10. In
the gas phase, there is a significant number of colloids located close to the wall, and density
profiles (not shown) indicate strong adsorption of colloids at both walls in the capillary gas.
This is due to the polymer-induced depletion attraction between colloid particle and wall. The
capillary liquid is dense in colloids and polymers tend to build clusters of overlapping particles.
We have simulated many such coexisting states for varying densities and values of H /σc.
The resulting set of phase diagrams is depicted in figure 4, where colloid packing fraction, ηc ,
and polymer reservoir packing fraction, ηpr , are taken as independent parameters. The binodals
possess a lower (in ηpr ) critical point, and a rapidly increasing density jump upon increasing
ηpr . The shape of the binodal is similar to that of the liquid–gas transition in simple fluids upon
identifying ηpr with inverse temperature.
Upon increasing confinement via decreasing H /σc, the critical point moves significantly
upwards to higher ηpr , again similar to the common trend of capillary-induced decrease of
critical temperature. The theoretical binodals agree well with the simulation data, except for a
slight overestimation of the coexisting liquid density, and a quite prominent underestimation
of ηpr at the critical point. The trend that the simulation result for the critical point is at higher
ηpr than in theory is already known in bulk [11], which we can confirm here. Decreasing H ,
the deviation gets stronger, and for H /σc = 2 quite strong deviations are found. As this is
already close to a two-dimensional situation, this finding is in accordance with the expectation
that the DFT becomes less accurate in the critical region.
Before discussing the ultimate crossover to two dimensions in more detail, we present the
same phase diagram in a different representation, namely as a function of colloid chemical
potential µc = ln(z c σc3 ) and of ηpr in figure 5. As a function of these variables, each pair of
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M Schmidt et al
Figure 3. Snapshots from computer simulation of the confined colloidal gas (left) in coexistence
with the confined colloidal liquid (right). Colloids (dark red) and polymers (light blue) are immersed
between two parallel plates (transparent; only the left-hand wall is shown) with separation distance
H /σc = 10 and orientation perpendicular to the horizontal axis. The polymer reservoir packing
fraction is ηpr = 1.116. Note that compared to figure 1 the system is rotated by 90◦ around the
viewing direction.
(This figure is in colour only in the electronic version)
3
H/σ c =2
3
5
10
bulk
2.5
η pr
2
1.5
1
0.5
0
0
0.05
0.1
0.15
ηc
0.2
0.25
0.3
0.35
Figure 4. Phase diagram of the AO model between parallel hard walls with separation distance
H /σc = 2, 3, 5, 10, ∞ (bulk) as a function of colloid packing fraction in the slit, ηc , and polymer
reservoir packing fraction, ηpr . DFT results for the binodal (curves; from top to bottom for increasing
H ; the dashed curve indicates the bulk result) and critical point (filled dots) are shown along with
MC results for coexisting states (symbols). Coexistence is along horizontal tie lines (not shown).
coexisting phases (gas and liquid phases with different ηc ) collapses to a single point as µc
and ηpr are equal for the coexisting phases. By reducing H , the critical point and the binodal
shifts to larger z c and larger ηpr . Consider a statepoint on the gas side of the phase diagram. If
this is chosen sufficiently close to the bulk binodal, it might well reside on the liquid side of
Capillary condensation of colloid–polymer mixtures confined between parallel plates
S3417
3
H/σ c =2
3
5
10
bulk
2.5
η pr
2
GAS
LIQUID
1.5
1
0.5
0
1
2
3
4
5
6
7
8
ln(zc σc3 )
Figure 5. The same as figure 4, but as a function of colloid chemical potential ln(z c σc3 ) and of
polymer reservoir packing fraction ηpr . Shown are the results for H /σc = 2, 3, 5, 10 (solid lines
and symbols as indicated from top to bottom) and bulk (dashed line and full squares).
the confined system, indicating a liquid phase in the capillary in equilibrium with a bulk gas.
Hence this demonstrates the occurrence of capillary condensation in the slit pore. Except for
the precise location of the critical point, the overall agreement between DFT and MC results
is quite satisfactory.
We have compared these findings to those predicted by the Kelvin equation, generalized
to binary mixtures [27]2 . Three statepoints were chosen: ηpr = 0.84 in the complete wetting
regime, ηpr = 1.1 just above the first layering transition [15], and ηpr = 2 deep in the partial
wetting regime, see figure 6. For the intermediate case the shift in ηpr and z c is correctly
predicted by the Kelvin equation, even for strong confinement. For high ηpr there are differences
emerging, especially for the cases H /σc = 2, 3. Close to the critical point, the performance is
not very satisfactory, as the growth of wetting layers at the walls is not taken into account.
We have attempted to include results for H very close to σc . As can be guessed
from the above results for larger H , we find that the values of ηpr needed to induce phase
separation become increasingly large and no longer fit on the same scale. In fact, close to
H /σc = 1 this is a trivial effect of dimensional crossover, which we can remedy by using
scaled variables. See figure 7 for the binodals in reservoir representation, but as a function
of ηc and ηpr (H − σc )/H . The simulation results for the quasi-two-dimensional system are
obtained by choosing H /σc = 1.01 (and we have checked the independence from this precise
value, by also considering H /σc = 1.001 at one statepoint and finding the same result within
the error bars). Indeed, the two-dimensional result is now on the same scale as the results for
larger slits. We also show the binodal obtained from the two-dimensional DFT [10], that is
equivalent to free volume theory [28] in two dimensions. Again the theory overestimates the
2
See equations (40) and (41) in [27]. Adapting those to the present case, we obtain the shifts in chemical
potentials, µi , of species i = c, p from simultaneous solution of 2(γwg − γwl )/(H − σ ) = µc ρcl + µp ρpl
g
g
g
and µp = −µc ρc /ρp , where ρil and ρi are the coexisting densities, γwl and γwg are the hard wall interface
tensions of the liquid and the gas phase, respectively, and σ ≡ σc = σp for the present size ratio, q = 1.
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3
2.5
GAS
ηpr
2
1.5
LIQUID
1
0.5
0
1
2
3
4
5
6
7
8
ln(z c σ c )
3
Figure 6. The same as figure 5; shown are the DFT results (curves) along with the predictions
from applying the generalized Kelvin equation (open symbols) for statepoints ηpr = 0.84, 1.1, 2
and H /σc = ∞ (bulk), 10, 5, 3, 2 (from right to left).
area of demixed states in the phase diagram, somewhat worse than in three dimensions. This
probably reflects the difficulty of predicting critical points in two dimensions reliably.
More remarkably, however, is that the scaling leads to an almost complete data collapse of
the MC results for all slit widths. This is an empirical finding, and we are not aware that such
a scaling has been found previously in simple liquids, where one had to relate temperature
to confinement in order to have a similar scaling. The binodals from DFT do not collapse as
nicely as the MC results; the scaling somewhat overcompensates the shift in ηpr with decreasing
H . Finally, in figure 8 we show the phase diagram as a function of two rescaled variables,
i.e. ηpr (H −σc )/H and ln(z c (H −σc )/H ) demonstrating that both fugacities need to be rescaled
to obtain the data collapse. However, in this representation the collapse is not as perfect as in
figure 7. Note also that the binodal for the 2D case lies on the opposite site of the bulk binodal,
as compared to the cases of finite H . This indicates non-monotonic behaviour in the range
1 < σc /H < 2. In fact the scaling overcompensates, as the binodals for 2 H /σc 10 are
in opposite order compared to figure 5.
5. Conclusions
In conclusion, we have considered a binary mixture of hard sphere colloidal particles and added
non-adsorbing polymer modelled by effective spheres in contact with a slit pore of parallel
hard walls. We find that this capillary induces stability of a (thin film) liquid phase that coexists
with the bulk colloidal gas phase. This behaviour can be understood from the strong depletion
attraction between (each) wall and the colloidal particles.
Both treatments, computer simulation and DFT, take into account all induced many-body
terms in the polymer-mediated effective interaction between the colloidal spheres. We achieve
this by treating, in both cases, the full binary mixture.
As an outlook on possible future work, we mention the interesting problem of freezing in
confined systems, and in particular how the pure hard sphere behaviour [29, 30] is changed by
adding polymer.
Capillary condensation of colloid–polymer mixtures confined between parallel plates
S3419
2
η p r (H- σc )/H
1.5
1
2
3
5
H/σc =10
bulk
2d
0.5
0
0
0.05
0.1
0.15
ηc
0.2
0.25
0.3
0.35
Figure 7. The same as figure 4, but as a function of ηc and the rescaled polymer packing fraction,
ηpr (H − σc )/H . Also shown are the results for the two-dimensional (2D) system obtained from
simulations for H /σc = 1.01 and free volume theory (short dashed curve).
2
GAS
η pr (H- σc )/H
1.5
LIQUID
1
H/σ c =2
3
5
10
bulk
2d
0.5
0
1
2
3
4
ln(z c σ c3
5
6
7
8
(H-σ c )/H)
Figure 8. The same as figure 5, but as a function of the rescaled variables ln(z c σc3 (H − σc )/H )
and ηpr (H − σc )/H .
We emphasize that our predictions can, in principle, be experimentally tested with colloid–
polymer mixtures confined between two glass walls. In particular, keeping this slit pore in
contact with a bulk colloidal gas, very close to coexistence, and monitoring the amount of
adsorbed colloid material in the slit should be fruitful.
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M Schmidt et al
Acknowledgments
We thank Dirk G A L Aarts, Henk N W Lekkerkerker, and Paul P F Wessels for many useful
and inspiring discussions. J M Brader is acknowledged for sending us his surface tension
data [15]. This work is part of the research program of the Stichting voor Fundamenteel
Onderzoek der Materie (FOM), that is financially supported by the Nederlandse Organisatie
voor Wetenschappelijk Onderzoek (NWO). Support by the DFG SFB TR6 ‘Physics of colloidal
dispersions in external fields’ is acknowledged.
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